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Midsphere Radius Of Snub Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ r_m = \frac{\sqrt{\frac{1}{1-0.94315125924}}}{2} \times \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{RA/V \times (((12 \times ((3\phi)+1)) \times (((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7) \times ((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

m⁻¹

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1. What is the Midsphere Radius of Snub Dodecahedron?

The midsphere radius of a snub dodecahedron is the radius of the sphere that is tangent to all edges of the polyhedron. It represents the sphere that fits perfectly "inside" the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the complex formula:

\[ r_m = \frac{\sqrt{\frac{1}{1-0.94315125924}}}{2} \times \frac{((20\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times 6 \times (3-((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)^{\frac{3}{2}})}{RA/V \times (((12 \times ((3\phi)+1)) \times (((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})^2)-(((36\phi)+7) \times ((\frac{\phi}{2}+\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}}+(\frac{\phi}{2}-\frac{\sqrt{\phi-\frac{5}{27}}}{2})^{\frac{1}{3}})))-((53\phi)+6))} \]

Where:

Explanation: This complex formula relates the midsphere radius to the surface-to-volume ratio using the mathematical properties of the snub dodecahedron and the golden ratio.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and materials science for understanding the spatial properties of the snub dodecahedron, which has applications in crystallography, nanotechnology, and architectural design.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a snub dodecahedron?
A: A snub dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices.

Q2: Why is the golden ratio (φ) used in this formula?
A: The golden ratio appears naturally in the geometry of the snub dodecahedron, making it essential for accurate mathematical descriptions of its properties.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the polyhedron. Smaller polyhedra have higher ratios, while larger ones have lower ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the snub dodecahedron. Other polyhedra have different mathematical relationships.

Q5: What precision can I expect from this calculation?
A: The calculator provides results with 6 decimal places of precision, which is sufficient for most geometric applications.

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